2011/06/04 by Quan‐Lin Li, Li, Quan-Lin · 2 citations
Business, Management and Accounting · Decision Sciences · Mathematics · #60-08 #60J #90-08 #94A #Advanced Queuing Theory Analysis #C.2 #C.4 #Classical Analysis and ODEs (math.CA) #FOS: Computer and information sciences #FOS: Mathematics #Networking and Internet Architecture (cs.NI) #Performance (cs.PF) #Probability (math.PR) #Probability and Risk Models #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.1106.0787
openalex publication_date 2011/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Marcel F. Neuts opened a key door in numerical computation of stochastic models by means of phase-type (PH) distributions and Markovian arrival processes (MAPs). To celebrate his 75th birthday, this paper reports a more general framework of Markovian supermarket models, including a system of differential equations for the fraction measure and a system of nonlinear equations for the fixed point. To understand this framework heuristically, this paper gives a detailed analysis for three important supermarket examples: M/G/1 type, GI/M/1 type and multiple choices, explains how to derive the system of differential equations by means of density-dependent jump Markov processes, and shows that the fixed point may be simply super-exponential through solving the system of nonlinear equations. Note that supermarket models are a class of complicated queueing systems and their analysis can not apply popular queueing theory, it is necessary in the study of supermarket models to summarize such a more general framework which enables us to focus on important research issues. On this line, this paper develops matrix-analytical methods of Markovian supermarket models. We hope this will be able to open a new avenue in performance evaluation of supermarket models by means of matrix-analytical methods.